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Answer :
To find out how many hours it will take for the population of bacteria to reach 3700, we start with the given exponential growth function:
[tex]\[ P(h) = 2900 e^{0.09h} \][/tex]
Here, [tex]\( P(h) \)[/tex] represents the number of bacteria after [tex]\( h \)[/tex] hours, and we want this to equal 3700:
[tex]\[ 3700 = 2900 e^{0.09h} \][/tex]
To solve for [tex]\( h \)[/tex], follow these steps:
1. Divide both sides by 2900 to isolate the exponential term:
[tex]\[ \frac{3700}{2900} = e^{0.09h} \][/tex]
2. Simplify the fraction:
[tex]\[ \frac{3700}{2900} \approx 1.27586 \][/tex]
So, the equation now looks like:
[tex]\[ 1.27586 = e^{0.09h} \][/tex]
3. Take the natural logarithm (ln) of both sides to solve for [tex]\( h \)[/tex]:
[tex]\[ \ln(1.27586) = \ln(e^{0.09h}) \][/tex]
Using the property of logarithms that [tex]\( \ln(e^x) = x \)[/tex], the equation simplifies to:
[tex]\[ \ln(1.27586) = 0.09h \][/tex]
4. Solve for [tex]\( h \)[/tex] by dividing both sides by 0.09:
[tex]\[ h = \frac{\ln(1.27586)}{0.09} \][/tex]
When evaluated, [tex]\( \ln(1.27586) \approx 0.2437 \)[/tex]. So:
[tex]\[ h \approx \frac{0.2437}{0.09} \][/tex]
[tex]\[ h \approx 2.7069 \][/tex]
Therefore, it will take approximately 2.71 hours for the number of bacteria to reach 3700.
[tex]\[ P(h) = 2900 e^{0.09h} \][/tex]
Here, [tex]\( P(h) \)[/tex] represents the number of bacteria after [tex]\( h \)[/tex] hours, and we want this to equal 3700:
[tex]\[ 3700 = 2900 e^{0.09h} \][/tex]
To solve for [tex]\( h \)[/tex], follow these steps:
1. Divide both sides by 2900 to isolate the exponential term:
[tex]\[ \frac{3700}{2900} = e^{0.09h} \][/tex]
2. Simplify the fraction:
[tex]\[ \frac{3700}{2900} \approx 1.27586 \][/tex]
So, the equation now looks like:
[tex]\[ 1.27586 = e^{0.09h} \][/tex]
3. Take the natural logarithm (ln) of both sides to solve for [tex]\( h \)[/tex]:
[tex]\[ \ln(1.27586) = \ln(e^{0.09h}) \][/tex]
Using the property of logarithms that [tex]\( \ln(e^x) = x \)[/tex], the equation simplifies to:
[tex]\[ \ln(1.27586) = 0.09h \][/tex]
4. Solve for [tex]\( h \)[/tex] by dividing both sides by 0.09:
[tex]\[ h = \frac{\ln(1.27586)}{0.09} \][/tex]
When evaluated, [tex]\( \ln(1.27586) \approx 0.2437 \)[/tex]. So:
[tex]\[ h \approx \frac{0.2437}{0.09} \][/tex]
[tex]\[ h \approx 2.7069 \][/tex]
Therefore, it will take approximately 2.71 hours for the number of bacteria to reach 3700.
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